Karteikarten
Sequences
Karten in diesem Stapel (21)
- 1inf seq definitionfunction whose domain is all +ve nos
- 2conv seq def{a(n)} converges to l if for every +ve no E, there is N such tht for all n>N , |a(n)-L|inf)a(n)=L where L is limit of seq
- 3seq div to +ve inffor every M there is N such that for all n>N a(n)>M
- 4seq div to -ve inffor every M there is N such that for all n>N a(n)
- 5prove sum, diff, const multi rulesimple
- 6prove product and quotient ruleuse boundedness
- 7boundedness def (above+below) GLB LUBthere is number M such that
1) a(n) - 8theorem 1 a (conv/div) seq of real nos are boundedconv
- 9and PROOFlet {an},{bn},{cn} be seq of real nos,
if a(n) <= b(n) <= c(n) holds fr all n beyond some index N,
and if lim(n->inf) a(n) = lim(n->inf) c(n) = L,
then lim(n->inf) b(n) = L also - 10if|b(n)|<= c(n) and c(n)->0 what happens to b(n)? by which thm?b(n)->0
sandwich - 11thm 3 continuous f(n) thm of seqif a(n)->L then f(an)->f(L)
- 12thm 4 L-Hospital which forms does it work on? PROOF0/0, inf/inf
if f(x) is defined fr all x>n(0) and {a(n)} is such that a=f(n) fr all n>n(0), then
lim(n->inf) f(n) = L implies {a(n)} -> L - 13COL PROOF (ln n)/n0
- 14COL PROOF n^(1/n)1
- 15COL PROOF x^(1/n)0
if x>0 - 16COL PROOF x^n0
if |x|<1
use E^(1/n)->1 - 17COL PROOF (1+x/n)^ne^x
log, lhospital - 18COL PROOF (x^n)/n!0
(|x|/M)^n->1 (cus M<|x|)
(M^M)/M! * above ->0
Sandwich - 19conv___boundedness (->, <-, <=>)->
others are nt always true - 20Monotonicityeither
non decreasing
non increasing - 21monotonic seq thm PROOFif a seq is both bounded and monotonic, then seq conv
use graph concept
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- university Flashcards on seq and series, created by Pushpita on 28/09/2015.
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